How do I halve the width of a mesh while keeping it parallel to other edges (locking to an axis doesn't work)?
You can change dimensions of a mesh by going to the Transform panel and input values directly into the fields under 'Scale'.

If you input 0.5 in the fields, assuming your mesh was already at Scale 1 for X, Y and Z, then it will be halved in all three dimensions while retaining it's current rectangular shape, it's position and alignment to other object and everything else. It will just become smaller.

You can also just change one or two of the dimensions and keep the other ones intact if you wish.